Usage

Arguments

Details

A Strauss/hard core process with interaction radius \(r\),
hard core distance \(h < r\), and
parameters \(\beta\) and \(\gamma\),
is a pairwise interaction point process
in which

distinct points are not allowed to come closer
than a distance \(h\) apart

each pair of points closer than \(r\) units apart
contributes a factor \(\gamma\) to the probability density.

This is a hybrid of the Strauss process and the hard core process.

The probability density is zero if any pair of points
is closer than \(h\) units apart, and otherwise equals
$$
f(x_1,\ldots,x_n) =
\alpha \beta^{n(x)} \gamma^{s(x)}
$$
where \(x_1,\ldots,x_n\) represent the
points of the pattern, \(n(x)\) is the number of points in the
pattern, \(s(x)\) is the number of distinct unordered pairs of
points that are closer than \(r\) units apart,
and \(\alpha\) is the normalising constant.

The interaction parameter \(\gamma\) may take any
positive value (unlike the case for the Strauss process).
If \(\gamma < 1\),
the model describes an ``ordered'' or ``inhibitive'' pattern.
If \(\gamma > 1\),
the model is ``ordered'' or ``inhibitive'' up to the distance
\(h\), but has an ``attraction'' between points lying at
distances in the range between \(h\) and \(r\).

The function ppm(), which fits point process models to
point pattern data, requires an argument
of class "interact" describing the interpoint interaction
structure of the model to be fitted.
The appropriate description of the Strauss/hard core process
pairwise interaction is
yielded by the function StraussHard(). See the examples below.

The canonical parameter \(\log(\gamma)\)
is estimated by ppm(), not fixed in
StraussHard().

If the hard core distance argument hc is missing or NA,
it will be estimated from the data when ppm is called.
The estimated value of hc is the minimum nearest neighbour distance
multiplied by \(n/(n+1)\), where \(n\) is the
number of data points.